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monoidal model category

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Monoidal categories

Contents

Idea

A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.

Definition

Definition

A (symmetric) monoidal model category is model category 𝒞\mathcal{C} equipped with the structure of a closed symmetric monoidal category (𝒞,,I)(\mathcal{C}, \otimes, I) such that the following two compatibility conditions are satisfied

  1. (pushout-product axiom) For every pair of cofibrations f:XYf \colon X \to Y and f:XYf' \colon X' \to Y', their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects

    (XY) XX(YX)YY, (X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \longrightarrow Y \otimes Y' \,,

    is itself a cofibration, which, furthermore, is acyclic if ff or ff' is.

    (Equivalently this says that the tensor product :C×CC\otimes : C \times C \to C is a left Quillen bifunctor.)

  2. (unit axiom) For every cofibrant object XX and every cofibrant resolution QIp II\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} I of the tensor unit II, the resulting morphism

    QIXp IXIXX Q I \otimes X \stackrel{p_I \otimes X}{\longrightarrow} I\otimes X \stackrel{\simeq}{\longrightarrow} X

is a weak equivalence.

Remark

The pushout-product axiom in def. 1 implies that for XX a cofibrant object, then the functor X()X \otimes (-) is preserves cofibrations and acyclic cofibrations.

In particular if the tensor unit II happens to be cofibrant, then the unit axiom in def. 1 is implied by the pushout-product axiom. (Because then QIIQ I \to I is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma. )

Definition

We say a monoidal model category, def. 1, satisfies the monoid axiom, def. 1, if every morphism that is obtained as a transfinite composition of pushouts of tensor products XfX\otimes f of acyclic cofibrations ff with any object XX is a weak equivalence.

(Schwede-Shipley 00, def. 3.3.).

Remark

In particular, the axiom in def. 2 says that for every object XX the functor X()X \otimes (-) sends acyclic cofibrations to weak equivalences.

Properties

Monoidal homotopy category

Abstractly

Proposition

For (𝒞,)(\mathcal{C},\otimes) a monoidal model category, def. 1, then the derived functor L\otimes^L of the tensor product makes the homotopy category of the model category itself into a monoidal category, such that the localization functor

γ:(𝒞,)(Ho(𝒞, L) \gamma \;\colon\; (\mathcal{C},\otimes) \longrightarrow (Ho(\mathcal{C}, \otimes^L)

is a lax monoidal functor.

Proof

Let VV be a monoidal model category, and consider it as a derivable category in the sense of (Shulman 11, section 8) with V QV_Q the subcategory of cofibrant objects and V R=VV_R=V. Then :V×VV\otimes :V\times V\to V is left derivable, i.e. it preserves the QQ-subcategories and weak equivalences. Since deriving is pseudofunctorial (and product-preserving) on the 2-category of derivable categories and left derivable functors, it follows immediately that Ho(V)Ho(V) is monoidal; this is (Shulman 11, example 8.13).

Now let V 0V_0 denote the category VV with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both QQ and RR. Then of course Ho(V 0)=VHo(V_0) = V, and V 0V_0 is also a pseudomonoid in derivable categories. The identity functor Id:V 0VId : V_0 \to V is not left derivable, since it does not preserve QQ-objects; but it is right derivable, since we took all objects in VV to be RR-objects (ignoring the fibrant objects in the model structure on VV). Of course IdId is strong monoidal, and this monoidality constraint can be expressed as a square in the double category of (Shulman 11) whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a monoidal functor may be expressed using products and double-categorical pasting in this double category. Therefore, it is all preserved by the double pseudofunctor HoHo; but Ho(Id)=γ:VHo(V)Ho(Id) = \gamma : V \to Ho(V). The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus γ\gamma is only lax monoidal.

Explicitly

Proposition

Let (𝒞,,1)(\mathcal{C}, \otimes, 1) be a monoidal model category with cofibrant tensor unit. Then the left derived functor L\otimes^L of the tensor product exists

𝒞 c×𝒞 c 𝒞 c γ×γ μ 1 γ Ho(𝒞)×Ho(𝒞) L Ho(𝒞) \array{ \mathcal{C}_c\times \mathcal{C}_c &\overset{\otimes}{\longrightarrow}& \mathcal{C}_c \\ {}^{\mathllap{\gamma \times \gamma}}\downarrow &\swArrow_{\mu^{-1}}& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C}) &\overset{\otimes^L}{\longrightarrow}& Ho(\mathcal{C}) }

and makes the homotopy category into a monoidal category (Ho(𝒞), L,γ(1))(Ho(\mathcal{C}), \otimes^L, \gamma(1)).

Moreover, the localization functor

γ:(𝒞 c,,1)(Ho(𝒞), L,γ(1)) \gamma \;\colon\; (\mathcal{C}_c, \otimes, 1) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(1))

on the category of cofibrant objects is a strong monoidal functor with structure morphism the inverse of the above natural isomorpmism

μ X,Y:γ(X) Lγ(Y)γ(XY). \mu_{X,Y} \;\colon\; \gamma(X)\otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y) \,.
Proof

For the left derived functor (def.) of the tensor product

𝒞×𝒞𝒞 \otimes \; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}

to exist, it is sufficient that its restriction to the subcategory

(𝒞×𝒞) c𝒞 c×𝒞 c (\mathcal{C} \times \mathcal{C})_c \simeq \mathcal{C}_c \times \mathcal{C}_c

of cofibrant objects preserves acyclic cofibrations (Ken Brown's lemma, here).

Every morphism (f,g)(f,g) in the product category 𝒞 c×𝒞 c\mathcal{C}_{c}\times \mathcal{C}_{c} may be written as a composite of a pairing with an identity morphisms

(f,g):(c 1,d 1)(id c 1,g)(c 1,d 2)(f,id c 2)(c 2,d 2). (f,g) \;\colon\; (c_1, d_1) \overset{(id_{c_1},g)}{\longrightarrow} (c_1,d_2) \overset{(f,id_{c_2})}{\longrightarrow} (c_2,d_2) \,.

Now since the pushout product (with respect to tensor product) with the initial morphism (*c 1)(\ast \to c_1) is equivalently the tensor product

(*c 1) gid c 1g (\ast \to c_1) \Box_{\otimes} g \;\simeq\; id_{c_1} \otimes g

and

f (*c 2)fid c 2 f \Box_{\otimes} (\ast \to c_2) \;\simeq\; f \otimes id_{c_2}

the pushout-product axiom (def. 1) implies that on the subcategory of cofibrant objects the functor \otimes preserves acyclic cofibrations. (This is why one speaks of a Quillen bifunctor, see also Hovey 99, prop. 4.3.1).

Hence L\otimes^L exists.

By the same decomposition and using the universal property of the localization of a category (def.) one finds that for 𝒞\mathcal{C} and 𝒟\mathcal{D} any two categories with weak equivalences (def.) then the localization of their product category is the product category of their localizations:

(𝒞×𝒟)[(W 𝒞W 𝒟) 1](𝒞[W 𝒞 1])×(𝒟[W 𝒟 1]). (\mathcal{C} \times \mathcal{D})[(W_{\mathcal{C}} \sqcup W_{\mathcal{D}})^{-1}] \simeq (\mathcal{C}[W^{-1}_{\mathcal{C}}]) \times (\mathcal{D}[W^{-1}_{\mathcal{D}}]) \,.

With this, the universal property as a localization (def.) of the homotopy category of a model category (thm.) induces associators: Let

𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 γ 𝒞×𝒞×𝒞 η γ 𝒞 Ho(𝒞)×Ho(𝒞)×Ho(𝒞) (() L()) L() Ho(𝒞),𝒞 c×𝒞 c×𝒞 c ()(()()) 𝒞 γ 𝒞×𝒞×𝒞 η γ 𝒞 Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞) \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\,,\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) }

be the natural isomorphism exhibiting the derived functors of the two possible tensor products of three objects, as shown at the top. By pasting the second with the associator natural isomorphism of 𝒞\mathcal{C} we obtain another such factorization for the first, as shown on the left below,

()𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 = α = 𝒞 c×𝒞 c×𝒞 c ()(()()) 𝒞 γ 𝒞×𝒞×𝒞 η γ 𝒞 Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞)𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 γ 𝒞×𝒞×𝒞 η γ 𝒞 Ho(𝒞)×Ho(𝒞)×Ho(𝒞) (() L()) L() Ho(𝒞) = α L = Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞) (\star) \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha}& \downarrow^{\mathrlap{=}} \\ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_\eta& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha^L}& \downarrow^{\mathrlap{=}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C}) &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}& Ho(\mathcal{C}) }

and hence by the universal property of the factorization through the derived functor, there exists a unique natural isomorphism α L\alpha^L such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the pentagon identity satisfied by α\alpha implies a pentagon identity for α L\alpha^L, and so α L\alpha^L is an associator for L\otimes^L.

The above equation on pasting composites of natural isomorphism is equivalently just the coherence law for a monoidal functor:

(γ(X) Lγ(Y)) Lγ(Z) α γ(X),γ(Y),γ(Z) L γ(X) L(γ(Y) Lγ(Z)) η η γ((XY)Z) γ(α) γ(X(YZ)). \array{ (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) &\overset{\alpha^L_{\gamma(X), \gamma(Y), \gamma(Z)}}{\longrightarrow}& \gamma(X) \otimes^L (\gamma(Y) \otimes^L \gamma(Z)) \\ {}^{\mathllap{\eta'}}\uparrow && \uparrow^{\mathrlap{\eta}} \\ \gamma( (X \otimes Y) \otimes Z ) &\underset{\gamma(\alpha)}{\longrightarrow}& \gamma(X \otimes (Y \otimes Z)) } \,.

Examples

Classical homotopy theory

Homological algebra and stable homotopy theory

With respect to a symmetric monoidal smash product of spectra:

(Schwede-Shipley 00, MMSS 00, theorem 12.1 (iii) with prop. 12.3)

The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.

Categorical model structures

Model structure on GG-objects

Assumption

Let \mathcal{E} be a category equipped with the structure of

such that

Proposition

Under these conditions there is for each finite group GG the structure of a monoidal model category on the category BG\mathcal{E}^{\mathbf{B}G} of objects in \mathcal{E} equipped with a GG-action, for which the forgetful functor

BG \mathcal{E}^{\mathbf{B}G} \to \mathcal{E}

preserves and reflects fibrations and weak equivalences.

See for instance (BergerMoerdijk 2.5).

Model structure on monoids

See model structure on monoids in a monoidal model category.

References

A textbook reference is

Some relevant homotopy category background is in

The monoidal structures for a symmetric monoidal smash product of spectra are due to

Further variation of the axiomatics is discussed in

The monoidal model structure on BG\mathcal{E}^{\mathbf{B}G} is discussed for instance in

Relation to symmetric monoidal (infinity,1)-categories is discussed in

Revised on July 4, 2016 06:50:24 by Urs Schreiber (131.220.184.222)